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Posterior Summary and Visualization Utilities

This vignette demonstrates the summary and plotting utilities available for stochtree models.

Setup

Load necessary packages

library(stochtree)
import numpy as np
import matplotlib.pyplot as plt
from stochtree import BARTModel, BCFModel, plot_parameter_trace

Set a seed for reproducibility

random_seed = 1234
set.seed(random_seed)
random_seed = 1234
rng = np.random.default_rng(random_seed)

Supervised Learning

We begin with the supervised learning use case served by the bart() function.

Below we simulate a simple regression dataset.

n <- 1000
p_x <- 10
p_w <- 1
X <- matrix(runif(n * p_x), ncol = p_x)
W <- matrix(runif(n * p_w), ncol = p_w)
f_XW <- (((0 <= X[, 10]) & (0.25 > X[, 10])) *
  (-7.5 * W[, 1]) +
  ((0.25 <= X[, 10]) & (0.5 > X[, 10])) * (-2.5 * W[, 1]) +
  ((0.5 <= X[, 10]) & (0.75 > X[, 10])) * (2.5 * W[, 1]) +
  ((0.75 <= X[, 10]) & (1 > X[, 10])) * (7.5 * W[, 1]))
noise_sd <- 1
y <- f_XW + rnorm(n, 0, 1) * noise_sd
n = 1000
p_x = 10
p_w = 1
X = rng.uniform(size=(n, p_x))
W = rng.uniform(size=(n, p_w))
# R uses X[,10] (1-indexed) = Python X[:,9]
f_XW = (
    ((X[:, 9] >= 0)    & (X[:, 9] < 0.25)) * (-7.5 * W[:, 0]) +
    ((X[:, 9] >= 0.25) & (X[:, 9] < 0.5))  * (-2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.5)  & (X[:, 9] < 0.75)) * ( 2.5 * W[:, 0]) +
    ((X[:, 9] >= 0.75) & (X[:, 9] < 1.0))  * ( 7.5 * W[:, 0])
)
noise_sd = 1.0
y = f_XW + rng.standard_normal(n) * noise_sd

Now we fit a simple BART model to the data.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3
)
bart_model <- stochtree::bart(
  X_train = X,
  y_train = y,
  leaf_basis_train = W,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)
bart_model = BARTModel()
bart_model.sample(
    X_train=X,
    y_train=y,
    leaf_basis_train=W,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3
    },
)

We obtain a high level summary of the BART model by running print().

print(bart_model)
stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
print(bart_model)
BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function.

summary(bart_model)
stochtree::bart() run with mean forest, global error variance model, and mean forest leaf scale model
Continuous outcome was modeled as Gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.884, standard deviation = 0.048, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.7932468 0.8223112 0.8519496 0.8824451 0.9139656 0.9469236 0.9840282 
Summary of leaf scale posterior: 
3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
       2.5%         10%         25%         50%         75%         90% 
0.004674266 0.005117139 0.005622974 0.006336524 0.007146933 0.008092304 
      97.5% 
0.009923685 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = -0.064, standard deviation = 3.285, quantiles:
      2.5%        10%        25%        50%        75%        90%      97.5% 
-6.8474814 -4.9268222 -1.8344594 -0.1213777  2.0122940  4.2990040  6.5187649 
print(bart_model.summary())
BART Model Summary:
-------------------
BARTModel run with mean forest, global error variance model, and mean forest leaf scale model
Outcome was modeled as gaussian with a leaf regression prior with 1 bases for the mean forest
Outcome was standardized
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.932, standard deviation = 0.049, quantiles:
    2.5%: 0.836
   10.0%: 0.870
   25.0%: 0.898
   50.0%: 0.931
   75.0%: 0.965
   90.0%: 0.995
   97.5%: 1.030
Summary of leaf scale posterior: 3000 samples, mean = 0.007, standard deviation = 0.001, quantiles:
    2.5%: 0.004
   10.0%: 0.005
   25.0%: 0.006
   50.0%: 0.006
   75.0%: 0.007
   90.0%: 0.008
   97.5%: 0.009
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 0.105, standard deviation = 3.287, quantiles:
    2.5%: -6.758
   10.0%: -4.319
   25.0%: -1.940
   50.0%: 0.170
   75.0%: 2.085
   90.0%: 4.411
   97.5%: 6.593

None

We can use the plot() function to produce a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

plot(bart_model)

ax = plot_parameter_trace(bart_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function to pull the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), leaf scale \(\sigma^2_{\ell}\), in-sample mean function predictions y_hat_train) and then plot any subset or transformation of these values.

y_hat_train_samples <- extractParameter(bart_model, "y_hat_train")
obs_index <- 1
plot(
  y_hat_train_samples[obs_index, ],
  type = "l",
  main = paste0("In-Sample Predictions Traceplot, Observation ", obs_index),
  xlab = "Index",
  ylab = "Parameter Values"
)

y_hat_train_samples = bart_model.extract_parameter("y_hat_train")
obs_index = 0
fig, ax = plt.subplots()
ax.plot(y_hat_train_samples[obs_index, :])
ax.set_title(f"In-Sample Predictions Traceplot, Observation {obs_index}")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
plt.show()

Causal Inference

We now run the same demo for the causal inference use case served by the bcf() function in R and the BCFModel Python class.

Below we simulate a simple dataset for a causal inference problem with binary treatment and continuous outcome.

# Generate covariates and treatment
n <- 1000
p_X = 5
X = matrix(runif(n * p_X), ncol = p_X)
pi_X = 0.25 + 0.5 * X[, 1]
Z = rbinom(n, 1, pi_X)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[, 3]
tau_X = X[, 2] * 2 - 1

# Generate outcome
epsilon = rnorm(n, 0, 1)
y = mu_X + tau_X * Z + epsilon
# Generate covariates and treatment
n = 1000
p_X = 5
X = rng.uniform(size=(n, p_X))
pi_X = 0.25 + 0.5 * X[:, 0]
Z = rng.binomial(1, pi_X, n).astype(float)

# Define the outcome mean functions (prognostic and treatment effects)
mu_X = pi_X * 5 + 2 * X[:, 2]
tau_X = X[:, 1] * 2 - 1

# Generate outcome
epsilon = rng.standard_normal(n)
y = mu_X + tau_X * Z + epsilon

Now we fit a simple BCF model to the data

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 1000
general_params <- list(
  num_threads = 1, 
  num_chains = 3, 
  adaptive_coding = TRUE
)
bcf_model <- stochtree::bcf(
  X_train = X,
  y_train = y,
  Z_train = Z,
  num_gfr = num_gfr,
  num_burnin = num_burnin,
  num_mcmc = num_mcmc,
  general_params = general_params
)
bcf_model = BCFModel()
bcf_model.sample(
    X_train=X,
    Z_train=Z,
    y_train=y,
    propensity_train=pi_X,
    num_gfr=10,
    num_burnin=0,
    num_mcmc=1000,
    general_params={
      "num_threads": 1, 
      "num_chains": 3, 
      "adaptive_coding": True
    },
)

We obtain a high level summary of the BCF model by running print().

print(bcf_model)
stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
print(bcf_model)
BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

For a more detailed summary (including the information above), we use the summary() function / method.

summary(bcf_model)
stochtree::bcf() run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
outcome was standardized
An internal propensity model was fit using stochtree::bart() in lieu of user-provided propensity scores
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning) 
Summary of sigma^2 posterior: 
3000 samples, mean = 0.944, standard deviation = 0.045, quantiles:
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.8610582 0.8871552 0.9143950 0.9427175 0.9728906 1.0035948 1.0341002 
Summary of prognostic forest leaf scale posterior: 
3000 samples, mean = 0.002, standard deviation = 0.000, quantiles:
        2.5%          10%          25%          50%          75%          90% 
0.0009099522 0.0010512771 0.0012100918 0.0014282627 0.0017172860 0.0020797403 
       97.5% 
0.0024886994 
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.481, mean (treated) = 0.999, standard deviation (control) = 0.352, standard deviation (treated) = 0.328
quantiles (control):
       2.5%         10%         25%         50%         75%         90% 
-1.18581860 -0.95360454 -0.71839480 -0.46956822 -0.23107997 -0.02999749 
      97.5% 
 0.14971456 
quantiles (treated):
     2.5%       10%       25%       50%       75%       90%     97.5% 
0.4296348 0.5923537 0.7681665 0.9715576 1.2103727 1.4362187 1.6930827 
Summary of treatment effect intercept (tau_0) posterior: 
3000 samples, mean = 0.093, standard deviation = 0.787, quantiles:
       2.5%         10%         25%         50%         75%         90% 
-1.51028936 -1.15052170 -0.33243786  0.08273915  0.77722859  1.09008786 
      97.5% 
 1.34300067 
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.487, standard deviation = 0.943, quantiles:
    2.5%      10%      25%      50%      75%      90%    97.5% 
1.683982 2.297164 2.805936 3.501198 4.131309 4.722664 5.403819 
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = 0.067, standard deviation = 0.512, quantiles:
        2.5%          10%          25%          50%          75%          90% 
-0.717762486 -0.548382932 -0.389178871 -0.004173154  0.563639161  0.760393172 
       97.5% 
 0.880757764 
print(bcf_model.summary())
BCF Model Summary:
------------------
BCFModel run with prognostic forest, treatment effect forest, global error variance model, prognostic forest leaf scale model, and treatment effect intercept model
Outcome was modeled as gaussian
Treatment was binary and its effect was estimated with adaptive coding
Outcome was standardized
User-provided propensity scores were included in the model
The sampler was run for 10 GFR iterations, with 3 chains of 0 burn-in iterations and 1000 MCMC iterations, retaining every iteration (i.e. no thinning)

Summary of sigma^2 posterior: 3000 samples, mean = 0.869, standard deviation = 0.041, quantiles:
    2.5%: 0.791
   10.0%: 0.816
   25.0%: 0.840
   50.0%: 0.869
   75.0%: 0.898
   90.0%: 0.921
   97.5%: 0.952
Summary of prognostic forest leaf scale posterior: 3000 samples, mean = 0.002, standard deviation = 0.000, quantiles:
    2.5%: 0.001
   10.0%: 0.001
   25.0%: 0.001
   50.0%: 0.002
   75.0%: 0.002
   90.0%: 0.002
   97.5%: 0.003
Summary of adaptive coding parameters: 
3000 samples, mean (control) = -0.538, mean (treated) = 1.131, standard deviation (control) = 0.315, standard deviation (treated) = 0.349
quantiles (control):
    2.5%: -1.198
   10.0%: -0.948
   25.0%: -0.745
   50.0%: -0.514
   75.0%: -0.309
   90.0%: -0.149
   97.5%: 0.010

quantiles (treated):
    2.5%: 0.505
   10.0%: 0.698
   25.0%: 0.891
   50.0%: 1.112
   75.0%: 1.343
   90.0%: 1.578
   97.5%: 1.884
Summary of treatment effect intercept (tau_0) posterior: 3000 samples, mean = -0.105, standard deviation = 0.665, quantiles:
    2.5%: -1.061
   10.0%: -0.837
   25.0%: -0.598
   50.0%: -0.212
   75.0%: 0.275
   90.0%: 0.665
   97.5%: 1.721
Summary of in-sample posterior mean predictions: 
1000 observations, mean = 3.481, standard deviation = 0.962, quantiles:
    2.5%: 1.859
   10.0%: 2.209
   25.0%: 2.769
   50.0%: 3.471
   75.0%: 4.161
   90.0%: 4.769
   97.5%: 5.330
Summary of in-sample posterior mean CATEs: 
1000 observations, mean = -0.028, standard deviation = 0.622, quantiles:
    2.5%: -1.077
   10.0%: -0.905
   25.0%: -0.574
   50.0%: 0.068
   75.0%: 0.540
   90.0%: 0.743
   97.5%: 0.911

None

In R, we have a plot() that produces a traceplot of model terms like the global error scale \(\sigma^2\) or (if \(\sigma^2\) is not sampled) the first observation of cached train set predictions.

In Python, we provide a plot_parameter_trace() function for requesting a traceplot of a specific model parameter.

plot(bcf_model)

ax = plot_parameter_trace(bcf_model, term="global_error_scale")
plt.show()

For finer-grained control over which parameters to plot, we can also use the extractParameter() function in R or the extract_parameter() method in Python to query the posterior distribution of any valid model term (e.g., global error scale \(\sigma^2\), prognostic forest leaf scale \(\sigma^2_{\mu}\), CATE forest leaf scale \(\sigma^2_{\tau}\), adaptive coding parameters \(b_0\) and \(b_1\) for binary treatment, in-sample mean function predictions y_hat_train, in-sample CATE function predictions tau_hat_train) and then plot any subset or transformation of these values.

adaptive_coding_samples <- extractParameter(bcf_model, "adaptive_coding")
plot(
  adaptive_coding_samples[1, ],
  type = "l",
  main = "Adaptive Coding Parameter Traceplot",
  xlab = "Index",
  ylab = "Parameter Values",
  ylim = range(adaptive_coding_samples),
  col = "blue"
)
lines(adaptive_coding_samples[2, ], col = "orange")
legend(
  "topright",
  legend = c("Control", "Treated"),
  lty = 1,
  col = c("blue", "orange")
)

adaptive_coding_samples = bcf_model.extract_parameter("adaptive_coding")
fig, ax = plt.subplots()
ax.plot(adaptive_coding_samples[0, :], color="blue", label="Control")
ax.plot(adaptive_coding_samples[1, :], color="orange", label="Treated")
ax.set_title("Adaptive Coding Parameter Traceplot")
ax.set_xlabel("Index")
ax.set_ylabel("Parameter Values")
ax.legend(loc="upper right")
plt.show()